Question

Cantor's Diagonal Proof Tutorial
Creator of Tutorial: malcolmmcswain

Answer 1

What is Cantor's Diagonal Proof?

Georg Cantor was a brilliant mathematician who invented all sorts of useful tools for categorizing the endlessly confusing thing called infinity. He even went so far as to say that there were different sizes of infinity. He said that the first kind of infinity was Aleph Null.
\[\aleph _{0}\]

Answer 2

Aleph Null was the countable infinity. Cantor said the natural numbers, (0,1,2,3,4,5…) the integers, (0,-1,1,-2,2-3,3…) and even the rational numbers (1/1,-1/2,2/3…) were all countable.

However, he said the REALS (a number set seemingly to the rationals, but including the pesky numbers like pi called irrational) were NOT COUNTABLE.

Answer 3

Well, of course he needed to prove this, so this is what he came up with.
Suppose, for instance, you WERE able to make a list of every single real number. This infinitely long list could be considered countable, right? Given an infinite amount of time, you should be able to list out every single real…
Wrong.

Let's take a look at this list again…

Answer 4

There is a way to build a number that will not be on this list.

If I take the first digit of the first number, and add one, then take the second digit of the second number and add one, then take the third digit of the third number, and add one (and so on forever) we build an infinite, non-repeating decimal that won't be on the list.

Answer 5

How? Well, it can't be the same as the first number, because the first digit of the created number is one more than the first digit of the first number on the list, and the second digit of the created number is one more than the second digit of the second number on the list, and so on…

Therefore, the reals are uncountable.

Answer 6

Cantor named this new kind of infinity Aleph 1,

\[\aleph _{1}\]
and his discovery revolutionized the way we think about infinity.

Answer 7

9 years ago whaaaaaaaaaaaaaaaaaaa